thomas.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
optim
to control the optimisation algorithm. See Details.pcf.ppp
to control the smoothing in the estimation of the
pair correlation function.pcf
. The argument X
can be either
[object Object],[object Object]
The algorithm fits the Thomas point process to X
,
by finding the parameters of the Thomas model
which give the closest match between the
theoretical pair correlation function of the Thomas process
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast
.
The Thomas point process is described in scale
. The
named vector of stating values can use either sigma2
($\sigma^2$) or scale
as the name of the second
component, but the latter is recommended for consistency with other
cluster models.
The theoretical pair correlation function of the Thomas process is $$g(r) = 1 + \frac 1 {4\pi \kappa \sigma^2} \exp(-\frac{r^2}{4\sigma^2})).$$ The theoretical intensity of the Thomas process is $\lambda = \kappa \mu$.
In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters $\kappa$ and $\sigma^2$. Then the remaining parameter $\mu$ is inferred from the estimated intensity $\lambda$.
If the argument lambda
is provided, then this is used
as the value of $\lambda$. Otherwise, if X
is a
point pattern, then $\lambda$
will be estimated from X
.
If X
is a summary statistic and lambda
is missing,
then the intensity $\lambda$ cannot be estimated, and
the parameter $\mu$ will be returned as NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The Thomas process can be simulated, using rThomas
.
Homogeneous or inhomogeneous Thomas process models can also
be fitted using the function kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
}
"minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
"fv"
)
containing the observed values of the summary statistic
(observed
) and the theoretical values of the summary
statistic computed from the fitted model parameters.
}
Thomas, M. (1949) A generalisation of Poisson's binomial limit for use in ecology. Biometrika 36, 18--25.
Waagepetersen, R. (2007)
An estimating function approach to inference for
inhomogeneous Neyman-Scott processes.
Biometrics 63, 252--258.
}
[object Object]
thomas.estK
mincontrast
,
pcf
,
rThomas
to simulate the fitted model.